Open Access

Strong Convergence of Monotone Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansive Mappings

Fixed Point Theory and Applications20092009:261932

DOI: 10.1155/2009/261932

Accepted: 21 September 2009

Published: 11 October 2009

Abstract

We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping in a Banach space by using monotone hybrid iteration method. By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Banach space.

1. Introduction

Let be a real Banach space and let be the dual space of Let be a maximal monotone operator from to . It is well known that many problems in nonlinear analysis and optimization can be formulated as follows. Find a point satisfying

(1.1)

We denote by the set of all points such that Such a problem contains numerous problems in economics, optimization, and physics and is connected with a variational inequality problem. It is well known that the variational inequalities are equivalent to the fixed point problems. There are many authors who studied the problem of finding a common element of the fixed point of nonlinear mappings and the set of solutions of a variational inequality in the framework of Hilbert spaces see; for instance, [111] and the reference therein.

A well-known method to solve problem (1.1) is called the proximal point algorithm: and

(1.2)

where and are the resovents of . Many researchers have studies this algorithm in a Hilbert space; see, for instance, [1215] and in a Banach space; see, for instance, [16, 17].

In 2005, Matsushita and Takahashi [18] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space : chosen arbitrarily,

(1.3)

where is the duality mapping on , . They proved that generated by (1.3) converges strongly to a fixed point of under condition that .

In 2008, Su et al. [19] modified the CQ method (1.3) for approximation a fixed point of a closed hemi-relatively nonexpansive mapping in a Banach space. Their method is known as the monotone hybrid method defined as the following. chosen arbitrarily, then

(1.4)

where is the duality mapping on , They proved that generated by (1.4) converges strongly to a fixed point of under condition that .

Note that the hybrid method iteration method presented by Matsushita and Takahashi [18] can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping.

Very recently, Inoue et al. [20] proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method.

Theorem 1.1 (Inoue et al. [20]).

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of Let be a monotone operator satisfying and let for all . Let be a relatively nonexpansive mapping such that . Let be a sequence generated by and
(1.5)

for all , where is the duality mapping on , and for some . If , then converges strongly to , where is the generalized projection from onto .

Employing the ideas of Inoue et al. [20] and Su et al. [19], we modify iterations (1.4) and (1.5) to obtain strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemi-relatively nonexpansive mappings in a Banach space. The results of this paper modify and improve the results of Inoue et al. [20], and some others.

2. Preliminaries

Throughout this paper, all linear spaces are real. Let and be the sets of all positive integers and real numbers, respectively. Let be a Banach space and let be the dual space of . For a sequence of and a point the weak convergence of to and the strong convergence of to are denoted by and , respectively.

Let be a Banach space. Then the duality mapping from into is defined by

(2.1)

Let be the unit sphere centered at the origin of . Then the space is said to be smooth if the limit

(2.2)

exists for all . It is also said to be uniformly smooth if the limit exists uniformly in . A Banach space is said to be strictly convex if whenever and . It is said to be uniformly convex if for each , there exists such that whenever and . We know the following (see, [21]):

(i)if in smooth, then is single valued;
1. (ii)

if is reflexive, then is onto;

2. (iii)

if is strictly convex, then is one to one;

(iv)if is strictly convex, then is strictly monotone;

(v)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

Let be a smooth strictly convex and reflexive Banach space and let be a closed convex subset of Throughout this paper, define the function by

(2.3)

Observe that, in a Hilbert space , (2.3) reduces to , for all . It is obvious from the definition of the function that for all ,

(1)

(2) ,

(3)

Following Alber [22], the generalized projection from onto is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem

(2.4)

Existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping In a Hilbert space, is the metric projection of onto .

Let be a closed convex subset of a Banach space and let be a mapping from into itself. We use to denote the set of fixed points of that is, Recall that a self-mapping is hemi-relatively nonexpansive if and for all and .

A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . We denote the set of all asymptotic fixed points of by . A hemi-relative nonexpansive mapping is said to be relatively nonexpansive if . The asymptotic behavior of a relatively nonexpansive mapping was studied in [23].

Recall that an operator in a Banach space is call closed, if and , then .

We need the following lemmas for the proof of our main results.

Lemma 2.1 (Kamimura and Takahashi [13]).

Let be a uniformly convex and smooth Banach space and let and be two sequences in such that either or is bounded. If , then .

Lemma 2.2 (Matsushita and Takahashi [18]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space and let be a relatively hemi-nonexpansive mapping from into itself. Then is closed and convex.

Lemma 2.3 (Alber [22], Kamimura and Takahashi [13]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space, and let . Then, if and only if for all .

Lemma 2.4 (Alber [22], Kamimura and Takahashi [13]).

Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space. Then
(2.5)

Let be a smooth, strictly convex, and reflexive Banach space, and let be a set-valued mapping from to with graph , domain and range We denote a set-valued operator from to by is said to be monotone of A monotone operator is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. It is known that a monotone mapping is maximal if and only if for for every implies that . We know that if is a maximal monotone operator, then is closed and convex; see [19] for more details. The following result is well known.

Lemma 2.5 (Rockafellar [24]).

Let be a smooth, strictly convex, and reflexive Banach space and let be a monotone operator. Then is maximal if and only if for all

Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying

(2.6)

Then we can define the resolvent by

(2.7)

We know that consists of one point. For , the Yosida approximation is defined by for all .

Lemma 2.6 (Kohsaka and Takahashi [25]).

Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of and let be a monotone operator satisfying
(2.8)

Let and let and be the resolvent and the Yosida approximation of , respectively. Then, the following hold:

(i)

(ii) ;

(iii)

Lemma 2.7 (Kamimura and Takahashi [13]).

Let be a uniformly convex and smooth Banach space and let . Then there exists a strictly increasing, continuous and convex function such that and
(2.9)

for all , where .

3. Main Results

In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space by using the monotone hybrid iteration method.

Theorem 3.1.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a monotone operator satisfying and let for all . Let be a closed hemi-relatively nonexpansive mapping such that . Let be a sequence generated by
(3.1)

for all , where is the duality mapping on and for some . If , then converges strongly to , where is the generalized projection from onto

Proof.

We first show that and are closed and convex for each . From the definition of and it is obvious that is closed and is closed and convex for each . Next, we prove that is convex.

Since
(3.2)
is equivalent to
(3.3)
which is affine in , and hence is convex. So, is a closed and convex subset of for all . Let Put for all Since and are hemi-relatively nonexpansive mappings, we have
(3.4)
So, for all , which implies that . Next, we show that for all . We prove that by induction. For , we have . Assume that for some . Because is the projection of onto by Lemma 2.3, we have
(3.5)
Since , we have
(3.6)
This together with definition of implies that and hence for all . So, we have that for all . This implies that is well defined. From definition of we have . So, from , we have
(3.7)
Therefore, is nondecreasing. It follows from Lemma 2.4 and that
(3.8)
for all . Therefore, is bounded. Moreover, by definition of , we know that and are bounded. So, the limit of exists. From we have that for any positive integer,
(3.9)
This implies that . Since is bounded, there exists such that . Using Lemma 2.7, we have, for with
(3.10)
where is a continuous, strictly increasing, and convex function with . Then the properties of the function yield that is a Cauchy sequence in . So there exists such that . In view of and definition of , we also have
(3.11)
It follows that Since is uniformly convex and smooth, we have from Lemma 2.1 that
(3.12)
So, we have Since is uniformly norm-to-norm continuous on bounded sets, we have
(3.13)
On the other hand, we have
(3.14)
This follows
(3.15)

From (3.13) and we obtain that

Since is uniformly norm-to-norm continuous on bounded sets, we have
(3.16)
From
(3.17)
we have
(3.18)
From (3.4), we have
(3.19)
Using and Lemma 2.6, we have
(3.20)
It follows that
(3.21)

From (3.13) and we have

Since is uniformly convex and smooth, we have from Lemma 2.1 that
(3.22)
From we have
(3.23)
Since and we have . Since is a closed operator and , is a fixed point of . Next, we show . Since is uniformly norm-to-norm continuous on bounded sets, from (3.22) we have
(3.24)
From , we have
(3.25)
Therefore, we have
(3.26)
For , from the monotonicity of , we have for all . Letting , we get . From the maximality of , we have . Finally, we prove that . From Lemma 2.4, we have
(3.27)
Since and we get from Lemma 2.4 that
(3.28)

By the definition of , it follows that and , whence . Therefore, it follows from the uniqueness of the that .

As direct consequences of Theorem 3.1, we can obtain the following corollaries.

Corollary 3.2.

Let be a uniformly convex and uniformly smooth Banach space. Let be a maximal monotone operator with and let for all . Let be a sequence generated by and
(3.29)

for all where is the duality mapping on , and for some . Then converges strongly to , where is the generalized projection from onto .

Proof.

Putting , and in Theorem 3.1, we obtain Corollary 3.2.

Let be a Banach space and let be a proper lower semicontinuous convex function. Define the subdifferential of as follows:

(3.30)

for each . Then, we know that is a maximal monotone operator; see [21] for more details.

Corollary 3.3 (Su et al. [19, Theorem  3.1]).

Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of and let be a closed hemi-relatively nonexpansive mapping from into itself such that . Let be a sequence generated by
(3.31)

for all , where is the duality mapping on and . If   then converges strongly to , where is the generalized projection from onto .

Proof.

Set in Theorem 3.1, where is the indicator function; that is,
(3.32)
Then, we have that is a maximal monotone operator and for , in fact, for any and , we have from Lemma 2.3 that
(3.33)

So, we obtain the desired result by using Theorem 3.1.

Since every relatively nonexpansive mapping is a hemi-relatively one, the following theorem is obtained directly from Theorem 3.1.

Theorem 3.4.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of Let be a monotone operator satisfying and let for all Let be a closed relatively nonexpansive mapping such that . Let be a sequence generated by
(3.34)

for all , where is the duality mapping on and for some . If then converges strongly to , where is the generalized projection from onto .

Corollary 3.5 (Su et al. [19, Theorem  3.2]).

Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of and let be a closed relatively nonexpansive mapping from into itself such that . Let be a sequence generated by
(3.35)

for all , where is the duality mapping on and . If   , then converges strongly to , where is the generalized projection from onto .

Proof.

Set in Theorem 3.4, where is the indicator function. So, from Theorem 3.4, we obtain the desired result.

Declarations

Acknowledgments

The authors would like to thank the referee for valuable suggestions that improve this manuscript and the Thailand Research Fund (RGJ Project) and Commission on Higher Education for their financial support during the preparation of this paper. The first author was supported by the Royal Golden Jubilee Grant PHD/0018/2550 and by the Graduate School, Chiang Mai University, Thailand.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University

References

1. Ceng L-C, Lee C, Yao J-C: Strong weak convergence theorems of implicit hybrid steepest-descent methods for variational inequalities. Taiwanese Journal of Mathematics 2008,12(1):227–244.
2. Ceng L-C, Ansari QH, Yao J-C: Viscosity approximation methods for generalized equilibrium problems and fixed point problems. Journal of Global Optimization 2009,43(4):487–502. 10.1007/s10898-008-9342-6
3. Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
4. Peng J-W, Yao J-C: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems. Mathematical and Computer Modelling 2009,49(9–10):1816–1828. 10.1016/j.mcm.2008.11.014
5. Peng J-W, Yao JC: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings. to appear in Taiwanese Journal of MathematicsGoogle Scholar
6. Peng J-W, Yao J-C: Some new extragradient-like methods for generalized equilibrium problems, fixed point problems and variational inequality problems. to appear in Optimization Methods and SoftwareGoogle Scholar
7. Peng J-W, Yao J-C: Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping. Journal of Global Optimization. In pressGoogle Scholar
8. Schaible S, Yao J-C, Zeng L-C: A proximal method for pseudomonotone type variational-like inequalities. Taiwanese Journal of Mathematics 2006,10(2):497–513.
9. Zeng L-C, Yao J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2006,10(5):1293–1303.
10. Zeng LC, Lin LJ, Yao JC: Auxiliary problem method for mixed variational-like inequalities. Taiwanese Journal of Mathematics 2006,10(2):515–529.
11. Zeng L-C, Wu S-Y, Yao J-C: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwanese Journal of Mathematics 2006,10(6):1497–1514.
12. Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. Journal of Approximation Theory 2000,106(2):226–240. 10.1006/jath.2000.3493
13. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002,13(3):938–945. 10.1137/S105262340139611X
14. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976,14(5):877–898. 10.1137/0314056
15. Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Mathematical Programming 2000,87(1):189–202.
16. Kamimura S, Kohsaka F, Takahashi W: Weak and strong convergence theorems for maximal monotone operators in a Banach space. Set-Valued Analysis 2004,12(4):417–429. 10.1007/s11228-004-8196-4
17. Kohsaka F, Takahashi W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstract and Applied Analysis 2004,2004(3):239–249. 10.1155/S1085337504309036
18. Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2005,134(2):257–266. 10.1016/j.jat.2005.02.007
19. Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, Article ID 284613, 2008:-8.Google Scholar
20. Inoue G, Takahashi W, Zembayashi K: Strong convergence theorems by hybrid methods for maximal monotone operator and relatively nonexpansive mappings in Banach spaces. to appear in Journal of Convex AnalysisGoogle Scholar
21. Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
22. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Katrosatos AG. Marcel Dekker, New York, NY, USA; 1996:15–50.Google Scholar
23. Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001,7(2):151–174. 10.1515/JAA.2001.151
24. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5
25. Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM Journal on Optimization 2008,19(2):824–835. 10.1137/070688717